Find extreme values for $f(x,y)=\sin^2(x)+\sin^2(y)$ over the constraint: $(x^2-y^2)^3+(x^2-y^2)=0$.

Question

Find extreme values for $f(x,y)=\sin^2(x)+\sin^2(y)$ over the constraint: $(x^2-y^2)^3+(x^2-y^2)=0$.

Im free to solve this problem with any method. So far I tried to solve this one using the method I understand better which is Lagrangian multipliers, this way for: $\nabla f= \lambda \nabla g$ where:

$$g(x,y)=(x^2-y^2)^3+(x^2-y^2).$$

Im pretty sure there is some shorter way to do this because proceeding the way I said I got pretty nightmarish equation systems to solve. I mean,

$$\nabla f= 2(\sin(x))\cos(x)i+ 2(\sin(y))\cos(y)j= \sin(2x)i+\cos(2y)j.$$

But

$$\nabla g =g_{x}i+g_{y}j=[6x^5-12x^3y^2+6xy^4+2x]i+ [-6x^4y+12x^2y^3-6y^5-2y]j.$$

So finding the values for $x,y$ and $\lambda$ such $\nabla f= \lambda \nabla g$ is satisfied becomes a nightmare as I said before. I would really aprecciate any help to understand and finish this problem. Thanks!

Answer 1

HINT:

$(x^2-y^2)^3+(x^2-y^2)=0 \iff (x^2-y^2)=0 \iff y=\pm x.$

Note that $f(x,-x)=f(x,x).$ Thus we have to find extreme values of $f(x,x)=2\sin^2 x.$

I am sure you can finish from here.